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<b>let </b><font color="Maroon">V</font> be  <a href="rusub_4.html#V1" target="_self">finite-dimensional</a> <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">W</font> be    <a href="rusub_1.html#M1">Subspace</a> of <font color="Maroon">V</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:20_1"/><i><font color="Green">E26</font></i>: 
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">W</font>
 ;<br/><b>consider </b><font color="Maroon">A</font> being    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">W</font>;<br/><b>consider </b><font color="Maroon">B</font> being    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E4:20_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="rusub_3.html#T24">RUSUB_3:24</a></i>;<br/><i><font color="Green">E29</font></i>:  <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font> = 
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font>
<b>by </b><i/>
;<br/><a NAME="E6:20_1"/>
( <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="rusub_1.html#D1">RUSUB_1:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>reconsider </b><font color="Maroon">B'</font> = <font color="Maroon">B</font> as  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> <b>by </b><i/>;<br/><div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:20_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">B</font>
 ;<br/><a NAME="E2:20_1_2"/><b>then </b>
<font color="Maroon">A</font> <a href="xboole_0.html#R2">c&lt;</a> <font color="Maroon">B</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D8">XBOOLE_0:def 8</a></i>;<br/><a NAME="E3:20_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <font color="Maroon">A'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="card_1.html#K4">card</a> <font color="Maroon">B'</font>
 <b>by </b><i><a class="ref" href="card_2.html#T67">CARD_2:67</a></i>;<br/><b>hence </b><a NAME="E4:20_1_2"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font> ;<br/><b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/> <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> = 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #)
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>
<br/>.= 
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">B</font>
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>
<br/>.= 
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="rusub_3.html#T28">RUSUB_3:28</a></i>
<br/>.= 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">W</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">W</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">W</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">W</font> #)
<b>by </b><i><a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>
<br/>.= 
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>
;<br/><b>hence </b><a NAME="E13:20_1"/>
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">W</font>
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:20"/>
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">W</font>
 ;<br/>

<b>then </b><i><font color="Green">E31</font></i>:  <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">V</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">V</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">V</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">V</font> #) = 
 <a href="rusub_1.html#K2">(Omega).</a> <font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>
<br/>.= 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">W</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">W</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">W</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">W</font> #)
<b>by </b><i><a class="ref" href="rusub_1.html#D3">RUSUB_1:def 3</a></i>
;<br/>
<b>consider </b><font color="Maroon">A</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font><b> such that </b><br/><a NAME="E5:20"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">A</font> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">V</font>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">B</font> being  <a href="finset_1.html#V1">finite</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font><b> such that </b><br/><a NAME="E7:20"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">B</font> is    <a href="rusub_3.html#M1">Basis</a> of <font color="Maroon">W</font>
 <b>by </b><i/>;<br/>
<a NAME="E8:20"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">A</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<a NAME="E9:20"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">B</font> is <a href="rlvect_3.html#V1">linearly-independent</a>
 
<b>by </b><i>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>;<br/>
<i><font color="Green">E40</font></i>:  <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">A</font> = 
 <a href="bhsp_1.html#G1">UNITSTR</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font>,the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">W</font>,the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">W</font>,the <a href="rlvect_1.html#U2">Mult</a> of <font color="Maroon">W</font>,the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">W</font> #)
<b>by </b><i>, , <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>
<br/>.= 
 <a href="rusub_3.html#K1">Lin</a> <font color="Maroon">B</font>
<b>by </b><i>, <a class="ref" href="rusub_3.html#D2">RUSUB_3:def 2</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">V</font> ;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">W</font> ;<br/>
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> = 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">A</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rusub_4.html#K1" target="_self">dim</a> <span class="p1">(<span class="default"><a href="rusub_3.html#K1">Lin</a> <font color="Maroon">B</font></span>)</span>
<b>by </b><i>, <a class="ref" href="rusub_4.html#T1" target="_self">Th1</a>, </i>
<br/>.= 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">B</font>
<b>by </b><i>, </i>
<br/>.= 
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">W</font>
<b>by </b><i>, </i>
;<br/>
<b>hence </b><a NAME="E14:20"/>
 <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="rusub_4.html#K1" target="_self">dim</a> <font color="Maroon">W</font>
 ;<br/>


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